**ROTATIONAL SYMMETRY**

An object has rotational symmetry if, when you rotate it by a certain amount, the resulting image is identical to the original image – or the image coincides with itself.

The ** angle of rotational symmetry** (

*or angle of symmetry)*is the smallest angle by which you can rotate the image so that you produce an identically oriented figure.

For example, consider an equilateral triangle. An equilateral triangle is a triangle where all sides are of equal length, and all angles measure 60°. The smallest angle by which we can rotate such a triangle to get an equilateral triangle of the same orientation is 120°; therefore, the angle of symmetry of an equilateral triangle is 120°.

Notice that we can also rotate the equilateral triangle by 240°and 360° to produce the same result!

The *order of rotational symmetry**(or order of rotation)* is the number of angles less than or equal to 360° by which we can rotate a shape to get an identically oriented shape.

It follows from the above diagram that the order of rotational symmetry of an equilateral triangle is 3 because there are exactly three angles less than or equal to 360° by which such a triangle can be rotated to coincide with itself (360°, 240° and 120°).

The order of rotational symmetry can be found by dividing 360 by the angle of rotational symmetry. In the triangle example: 360° ÷ 120° = 3.

**REFLECTIVE SYMMETRY**

An object is said to have reflective symmetry when a line can be drawn through some part of it, and the shapes on either side of the line are mirror images of each other. That is, if you placed a mirror on this line, and looked at the first half of the shape through the mirror, the resulting image would be identical to the second half.

The *line of reflection* is the line that divides these two mirror-image shapes. If you were to fold the image along the line of reflection, both sides would align perfectly.

For example, consider this butterfly:

The butterfly has reflective symmetry, because we can draw the dotted line through a part of the butterfly and the resulting pieces are mirror images of each other.

The ** total order of symmetry** of an object is the sum of the object’s order of rotational symmetry and its number of lines of reflection.

Mandalas exhibit a lot of rotational and reflective symmetry.

Consider the mandala below in applying the concepts that we have just discussed: angle of rotation, order of rotation, lines of symmetry, and total order of symmetry.

This mandala’s angle of rotation is 45°. The smallest angle by which we can rotate the mandala to produce an identical configuration is 45°.

Its order of rotation is 360° ÷ 45° = 8. The number of angles less than or equal to 360° by which we can rotate the mandala to get the same image is 8.

We can also see that this mandala has 8 lines of reflection. Each of these 8 lines divide the mandala into mirror image halves.

Therefore, the mandala’s total order of symmetry is equal to 8 + 8 = 16.

Pretty straight-forward!